Optimal charged membranes for the pressure-driven separations of ions of different mobilities: theoretical analysis

In the previous publication [A.E. Yaroshchuk, Asymptotic behavior in the pressure-driven separations of ions of different mobilities in charged porous membranes, J. Membr. Sci.], it has been found that strong separations of counterions of different mobilities (but the same charge) occur only at moderate to large membrane Péclet numbers. At realistic transmembrane pressure differences, the mode of moderate to large membrane Péclet numbers occurs only for membranes with relatively large pores. For them, the applicability of widely used homogeneous approximation becomes disputable. Therefore, the heterogeneity of ion distribution inside pores has been taken into account for the model of straight cylindrical capillaries. The account for microheterogeneity by means of non-linearized Poisson–Boltzmann equation has revealed the existence of optimal fixed charge densities as well as of optimal pore sizes. The latter are rather large (say, 8 nm for decinormal feed solutions). The active layer thickness of membranes with such pore sizes has to be essentially larger than in conventional NF membranes to diminish concentrational polarization. Thus, ‘optimal’ charged membranes for the pressure-driven separation of ions of different mobilities are quite different from conventional NF membranes. The latter are fine-porous, thin and only slightly charged while the former should be relatively coarse-porous, thick and have a large fixed charge density. To be effective, that fixed charge should be compensated by counterions involved in convective solution transfer (the so called electrokinetic charge). The existence of an extended hydrodynamically immobile layer near the pore surface noticeably diminishes the membrane separation efficiency. That is especially so if ions retain their electo-diffusional mobility within that layer.